Key Concepts: Geometric Progression (G.P.) and Binomial Theorem

This problem elegantly combines two fundamental mathematical concepts:

1. Geometric Progression (G.P.): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The sum of the first $n$ terms of a G.P. is given by the formula: $S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{where } r \neq 1$ Here, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

2. Binomial Theorem: This theorem provides a formula for expanding expressions of the form $(a+b)^n$. The general term (or $(k+1)^{th}$ term) in the expansion of $(1+x)^n$ is given by: $T_{k+1} = \binom{n}{k} x^k$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient, representing the coefficient of $x^k$.

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Step-by-Step Solution

1. Identify the Expression as a Geometric Progression (G.P.)

The given expression is: $(1 + x)^{10} + x(1 + x)^9 + x^2 (1 + x)^8 + \cdots + x^{10}$

We begin by carefully examining the structure of the terms. Let's list the first few terms and the last term:

• $T_1 = (1+x)^{10}$
• $T_2 = x(1+x)^9$
• $T_3 = x^2(1+x)^8$
• ...
• $T_{11} = x^{10}$ (This can be expressed as $x^{10}(1+x)^0$ for consistency)

Why this step? Recognizing this pattern as a standard series type, specifically a Geometric Progression, is the most crucial step. It allows us to use a powerful simplification formula instead of attempting to expand each individual term, which would be exceedingly complex and time-consuming.

To confirm it's a G.P., we check if the ratio of consecutive terms is constant. This constant ratio is called the common ratio ($r$).

• Calculate the ratio of the second term to the first term: $r = \frac{T_2}{T_1} = \frac{x(1+x)^9}{(1+x)^{10}} = \frac{x}{1+x}$
• Calculate the ratio of the third term to the second term (to verify consistency): $r = \frac{T_3}{T_2} = \frac{x^2(1+x)^8}{x(1+x)^9} = \frac{x}{1+x}$ Since the ratio is constant, the given expression is indeed a Geometric Progression.

From the series, we can identify its key components:

• First term ($a$): $(1+x)^{10}$
• Common ratio ($r$): $\frac{x}{1+x}$
• Number of terms ($n$): The powers of $x$ in the numerator increase from $x^0$ (in the first term) to $x^{10}$ (in the last term). This indicates there are $10 - 0 + 1 = 11$ terms.

2. Calculate the Sum of the G.P.

Now that we have identified the G.P. and its parameters, we can apply the sum formula for a Geometric Progression: $S_n = \frac{a(1 - r^n)}{1 - r}$.

Why this step? Directly expanding and adding 11 binomial expressions to find the coefficient of $x^7$ would be a monumental task. By using the G.P. sum formula, we consolidate the entire expression into a much simpler form, which makes finding the required coefficient straightforward.

Substitute the values of $a = (1+x)^{10}$, $r = \frac{x}{1+x}$, and $n=11$ into the sum formula: $S_{11} = \frac{(1+x)^{10} \left(1 - \left(\frac{x}{1+x}\right)^{11}\right)}{1 - \frac{x}{1+x}}$

First, let's simplify the denominator: $1 - \frac{x}{1+x} = \frac{(1+x) - x}{1+x} = \frac{1}{1+x}$

Now, substitute this simplified denominator back into the sum expression: $S_{11} = \frac{(1+x)^{10} \left(1 - \frac{x^{11}}{(1+x)^{11}}\right)}{\frac{1}{1+x}}$

To simplify further, multiply the numerator by the reciprocal of the denominator, and combine the terms inside the parenthesis: $S_{11} = (1+x)^{10} \cdot (1+x) \cdot \left(\frac{(1+x)^{11} - x^{11}}{(1+x)^{11}}\right)$ Combining the $(1+x)$ terms in the numerator: $S_{11} = (1+x)^{11} \cdot \frac{(1+x)^{11} - x^{11}}{(1+x)^{11}}$ Notice that the $(1+x)^{11}$ term in the numerator cancels with the $(1+x)^{11}$ term in the denominator (from the fraction within the parenthesis). This leaves us with the greatly simplified sum: $S_{11} = (1+x)^{11} - x^{11}$

3. Find the Coefficient of $x^7$ in the Sum

Our goal is to find the coefficient of $x^7$ in the simplified expression: $S_{11} = (1+x)^{11} - x^{11}$.

Why this step? With the expression simplified, we can now easily apply the Binomial Theorem to extract the coefficient of the desired power of $x$.

We will consider each term of the sum separately:

• For the term $-x^{11}$: The lowest power of $x$ in this term is $x^{11}$. Since $7 < 11$, there is no $x^7$ term in $-x^{11}$. Therefore, the coefficient of $x^7$ in $-x^{11}$ is $0$.

• For the term $(1+x)^{11}$: We use the Binomial Theorem. The general term containing $x^k$ in the expansion of $(1+x)^n$ is $\binom{n}{k} x^k$. In our case, $n=11$, and we are looking for the coefficient of $x^7$, so we set $k=7$. The coefficient of $x^7$ in $(1+x)^{11}$ is $\binom{11}{7}$.

Now, we calculate the binomial coefficient $\binom{11}{7}$: $\binom{11}{7} = \frac{11!}{7!(11-7)!} = \frac{11!}{7!4!}$

Tip for calculation: A helpful property of binomial coefficients is $\binom{n}{k} = \binom{n}{n-k}$. This means $\binom{11}{7} = \binom{11}{11-7} = \binom{11}{4}$. Calculating $\binom{11}{4}$ is often quicker as it involves fewer terms in the denominator.

Let's calculate $\binom{11}{4}$: $\binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1}$ We can simplify by canceling terms: $= 11 \times \frac{10}{2 \times 5} \times \frac{9}{3} \times \frac{8}{4}$ $= 11 \times 5 \times 3 \times 2$ $= 330$

So, the coefficient of $x^7$ in $(1+x)^{11}$ is $330$.

Finally, we combine the coefficients from both parts of the sum: Coefficient of $x^7$ in $[(1+x)^{11} - x^{11}]$ $= (\text{Coefficient of } x^7 \text{ in } (1+x)^{11}) + (\text{Coefficient of } x^7 \text{ in } -x^{11})$ $= 330 + 0$ $= 330$

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Common Mistakes to Avoid:

• Miscounting the number of terms ($n$) in the G.P.: Always be precise when determining $n$. If powers start from $0$ and go up to $k$, there are $k+1$ terms.
• Algebraic errors during G.P. sum simplification: Be very careful with combining fractions and exponent rules. A single mistake here will lead to an incorrect final expression.
• Overlooking parts of the simplified expression: If the final sum has multiple terms (e.g., $A - B$), ensure you check for the required coefficient in all terms. In this case, $-x^{11}$ did not contribute, but it's important to explicitly verify.
• Calculation errors for binomial coefficients: Double-check your arithmetic, or use the $\binom{n}{k} = \binom{n}{n-k}$ property to make calculations simpler.
• Discrepancy with given options: In this particular problem, the provided "Correct Answer: A (120)" conflicts with the result derived from the standard mathematical methods (330). In such situations, it's crucial to trust your logical derivations and calculations if you are confident in your application of the principles. Often, a question might ask for the coefficient in just the first term, which would be $\binom{10}{7} = 120$, but here it's clearly the sum.

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Summary and Key Takeaway

This problem is an excellent example of how to tackle seemingly complex series by first identifying underlying patterns. By recognizing the expression as a Geometric Progression, we could use its sum formula to simplify it to $(1+x)^{11} - x^{11}$. From this simpler form, finding the coefficient of $x^7$ was a direct application of the Binomial Theorem. The core takeaway is to always look for ways to simplify expressions using known formulas (like G.P. sum) before resorting to tedious direct expansion, and then meticulously apply the Binomial Theorem to find the specific coefficient. The final calculated coefficient of $x^7$ is $330$.